a For what values of k does the function ycos kt satisfy the

a.) For what values of k does the function y=cos kt satisfy the differential equation 4y^ll= -25y?

b.) For those values of k, verify that every member of of the family of functions y=A sin kt + B Cos kt is also a solution.

Solution

a) 4y\" = -25y -4*k^2*coskt = -25*coskt => 4k^2 = 25 k = 2.5 or -2.5 b) i) k = 2.5 y = Asin(2.5t) + Bcos(2.5t) 4y\" = -25y -4*2.5^2*(Asin2.5t + Bcos2.5t) = -25*(Asin2.5t + Bcos2.5t) -25*(Asin2.5t + Bcos2.5t) = -25*(Asin2.5t + Bcos2.5t) ii) k = -2.5 y = Asin(-2.5t) + Bcos(-2.5t) 4y\" = -25y -4*(-2.5)^2*(Asin(-2.5t) + Bcos(-2.5t)) = -25*(Asin(-2.5t) + Bcos(-2.5t)) -25*(Asin(-2.5t) + Bcos(-2.5t)) = -25*(Asin(-2.5t) + Bcos(-2.5t)) Hence, y=A sin kt + B Cos kt is also a solution for k = +- 2.5